(a) Determine k that renders f(x) a valid density func- tion. (b) Find the probability that a random error in mea- surement is less than 1/2. (c) For this particular measurement, it is undesirable if the magnitude of the error (i.e., [x]) exceeds 0.8. What is the probability that this occurs?
(a) Determine k that renders f(x) a valid density func- tion. (b) Find the probability that a random error in mea- surement is less than 1/2. (c) For this particular measurement, it is undesirable if the magnitude of the error (i.e., [x]) exceeds 0.8. What is the probability that this occurs?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![### Problem 3.30
Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error, and statisticians spend a great deal of time modeling these errors. Suppose the measurement error \( X \) of a certain physical quantity is decided by the density function
\[
f(x) =
\begin{cases}
k(3 - x^2), & -1 \leq x \leq 1, \\
0, & \text{elsewhere}.
\end{cases}
\]
**(a)** Determine \( k \) that renders \( f(x) \) a valid density function.
**(b)** Find the probability that a random error in measurement is less than \( \frac{1}{2} \).
**(c)** For this particular measurement, it is undesirable if the magnitude of the error (i.e., \(|x|\)) exceeds 0.8. What is the probability that this occurs?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b3f294e-d3c8-4475-8e8e-f2b525027412%2Fddb8b465-c9ab-4420-bcfb-2c88072516fc%2F2x1mz9d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 3.30
Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error, and statisticians spend a great deal of time modeling these errors. Suppose the measurement error \( X \) of a certain physical quantity is decided by the density function
\[
f(x) =
\begin{cases}
k(3 - x^2), & -1 \leq x \leq 1, \\
0, & \text{elsewhere}.
\end{cases}
\]
**(a)** Determine \( k \) that renders \( f(x) \) a valid density function.
**(b)** Find the probability that a random error in measurement is less than \( \frac{1}{2} \).
**(c)** For this particular measurement, it is undesirable if the magnitude of the error (i.e., \(|x|\)) exceeds 0.8. What is the probability that this occurs?
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